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چکیده : 

P-Splines are commonly used for derivative estimation where a non-linear relationship

exists between the response and explanatory variables. However, questions about the error

of these estimates have arisen. Incorporating an extra penalty term in a P-Spline model

is proposed as an improvement when derivative estimation is of primary concern. This

additive penalty approach to derivative estimation is shown to improve on the P-Spline

estimates based on the results of a simulation study to compare the performance when

estimating the first and second derivatives of six simulated functions. A method for

generating variability bands for P-Spline derivative estimates with and without an additive

penalty is given. The proposed additive penalty variability bands are shown to behave

better than their single penalty counterpart. Motivating examples in environmental and

sports science are used to demonstrate the need for accurate derivative estimates and the

benefit of using an additional penalty term to this end.